| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.SmoothTensorCurve |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.TensorEnergy |
| open Set Function |
| open scoped Classical BigOperators RealInnerProductSpace |
|
|
| def flowJetMajorant (A : ℕ → ℝ) (hA : ∀k,0≤A k) (n : ℕ) : ℝ := |
| (smoothTensorCurve_bounds A hA n).choose |
| lemma flowJetMajorant_nonneg (A : ℕ → ℝ) (hA : ∀k,0≤A k) (n : ℕ) : |
| 0≤flowJetMajorant A hA n := (smoothTensorCurve_bounds A hA n).choose_spec.1 |
|
|
| def flowCorrectionMajorant (A : ℕ → ℝ) (hA : ∀k,0≤A k) (n : ℕ) : ℝ := |
| ∑c : OrderedFinpartition n,A c.length*∏i,flowJetMajorant A hA (c.partSize i) |
| lemma flowCorrectionMajorant_nonneg (A : ℕ → ℝ) (hA : ∀k,0≤A k) (n : ℕ) : |
| 0≤flowCorrectionMajorant A hA n := |
| Finset.sum_nonneg (fun _ _ => mul_nonneg (hA _) (Finset.prod_nonneg (fun _ _ => flowJetMajorant_nonneg A hA _))) |
|
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| lemma multilinearTensor_sub {d n : ℕ} (f g : Point d → Point d) |
| (hf : ContDiff ℝ (⊤:ℕ∞) f) (hg : ContDiff ℝ (⊤:ℕ∞) g) (y : Point d) : |
| multilinearTensor (iteratedFDeriv ℝ n (fun z => f z-g z) y)= |
| multilinearTensor (iteratedFDeriv ℝ n f y)-multilinearTensor (iteratedFDeriv ℝ n g y) := by |
| change multilinearTensor (iteratedFDeriv ℝ n (f-g) y)=_ |
| rw [iteratedFDeriv_sub_apply (hf.contDiffAt.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))) |
| (hg.contDiffAt.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤)))] |
| ext c |
| simp only [multilinearTensor,sub_apply,inner_sub_right,Pi.sub_apply] |
|
|
| theorem smoothTensorCurve_correction {d n : ℕ} (hn : 0<n) (A : ℕ → ℝ) (hA : ∀k,0≤A k) |
| {R b η : ℝ} (hR : 0<R) (hR1 : R≤1) (hb : 0≤b) (hb1 : b≤1) (hη : 0≤η) |
| (X : ℝ × Point d → Point d) (V : ℝ → Point d → Point d) |
| (hX : SmoothTensorCurve d A R b X V) |
| (hv : ∀k u,u∈Icc 0 b → ∀z,AllSplitBound |
| (multilinearTensor (iteratedFDeriv ℝ k (V u) z)) (η*A k/R^(k-1))) |
| {t : ℝ} (ht : t∈Icc 0 b) (y : Point d) : |
| AllSplitBound (multilinearTensor (iteratedFDeriv ℝ n (fun z => X (t,z)-z) y)) |
| (η*flowCorrectionMajorant A hA n/R^(n-1)) := by |
| let T (u : ℝ) := multilinearTensor (iteratedFDeriv ℝ n (fun z => X (u,z)) y)- |
| multilinearTensor (iteratedFDeriv ℝ n (id : Point d → Point d) y) |
| let D (u : ℝ) := multilinearTensor (iteratedFDeriv ℝ n (fun z => V u (X (u,z))) y) |
| let M (k : ℕ) := (η*A k)/R^(k-1) |
| let N (k : ℕ) := flowJetMajorant A hA k/R^(k-1) |
| have hM (k : ℕ) : 0≤M k := by exact div_nonneg (mul_nonneg hη (hA k)) (pow_nonneg hR.le _) |
| have hN (k : ℕ) : 0≤N k := div_nonneg (flowJetMajorant_nonneg A hA _) (pow_nonneg hR.le _) |
| have hflow (k : ℕ) (u : ℝ) (hu : u∈Icc 0 b) : |
| AllSplitBound (multilinearTensor (iteratedFDeriv ℝ k (fun z => X (u,z)) y)) (N k) := |
| (smoothTensorCurve_bounds A hA k).choose_spec.2 d R b X V hR hR1 hb hb1 hX u hu y |
| have hbound (u : ℝ) (hu : u∈Ico 0 b) : |
| AllSplitBound (D u) (η*flowCorrectionMajorant A hA n/R^(n-1)) := by |
| have hs : ContDiff ℝ (⊤:ℕ∞) (fun z => X (u,z)) := hX.smooth.comp (contDiff_const.prodMk contDiff_id) |
| have hvs := (hX.velocity_smooth u ⟨hu.1,hu.2.le⟩).contDiffAt (x:=X (u,y)) |
| have he := iteratedFDeriv_comp_allSplit y |
| (hs.contDiffAt.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))) |
| (hvs.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))) M N hM hN |
| (fun k hk => hv k u ⟨hu.1,hu.2.le⟩ (X (u,y))) |
| (fun k hk => hflow k u ⟨hu.1,hu.2.le⟩) |
| have heq : (∑c : OrderedFinpartition n,M c.length*∏i,N (c.partSize i))= |
| η*flowCorrectionMajorant A hA n/R^(n-1) := by |
| dsimp only [M,N] |
| rw [partition_sum_scale hn (fun k => η*A k) (flowJetMajorant A hA) R] |
| simp only [flowCorrectionMajorant,←Finset.mul_sum,mul_assoc] |
| rw [heq] at he |
| exact he |
| have hinit : AllSplitBound (T 0) 0 := by |
| have he : (fun z => X (0,z))=(id : Point d → Point d) := funext hX.initial |
| dsimp only [T] |
| rw [he,sub_self] |
| exact allSplitBound_zero |
| have hpos : 0≤η*flowCorrectionMajorant A hA n/R^(n-1) := |
| div_nonneg (mul_nonneg hη (flowCorrectionMajorant_nonneg A hA n)) (pow_nonneg hR.le _) |
| have hg := allSplit_gronwall (T:=T) (D:=D) (K:=0) |
| ((spatialTensor_continuous hX.smooth y).sub continuous_const).continuousOn |
| (fun u hu => ((spatialTensor_time_deriv hX.smooth (V u) u |
| (hX.equation u ⟨hu.1,hu.2.le⟩) y).sub_const _).hasDerivWithinAt) |
| (le_refl (0:ℝ)) hpos (le_refl (0:ℝ)) hinit |
| (fun u hu => by simpa only [zero_mul,zero_add] using hbound u hu) ht |
| simp only [gronwallBound_K0,zero_add,sub_zero] at hg |
| have hfinal := hg.mono (mul_nonneg hpos ht.1) (mul_le_of_le_one_right hpos (ht.2.trans hb1)) |
| have hs : ContDiff ℝ (⊤:ℕ∞) (fun z => X (t,z)) := hX.smooth.comp (contDiff_const.prodMk contDiff_id) |
| rw [multilinearTensor_sub (fun z => X (t,z)) (fun z => z) hs contDiff_id] |
| exact hfinal |
| end LogConcaveSampling.TensorEnergy |
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| end |
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| end |
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| end |
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| end OAI |
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